Download Lecture 33

PHYS16 – Lecture 33
On a windy day in 1735, a new wig gives Bernoulli an idea.
Fluids: Bernoulli’s Principle
Fluids pre-question
• You are watering some flowers with a garden hose. You
want to water some flowers in the very back of the
garden. Assume that the flowers you want to water are
at the same level as the hose. What do you have to do
to the opening of the hose to increase the range by 4?
A) Decrease the area by a factor of 16
B) Decrease the area by a factor of 4
C) Decrease the area by a factor of 2
D) Increase the area by a factor of 2
E) Increase the area by a factor of 4
Outline for Fluids
• Fluid Statics
– Pressure and Pascal’s Principle
– Buoyant Force and Archimedes’ Principle
• Fluid Dynamics
– Ideal Fluids
– Equation of Continuity
– Bernoulli’s Equation
Buoyant Force and anchors…
• If I have a boat in a pond and I toss out an
anchor what happens to the water level in the
pond?
Fluid Dynamics: Ideal Fluids
Ideal Fluids
•
•
•
•
Incompressible – density is a constant
Nonviscous – ignore frictional effects
Irrotational – doesn’t rotate
Laminar – no acceleration
Streamlines represent fluid flow
Ideal Fluids
•
•
•
•
Mass is conserved
Energy is conserved
Momentum is conserved
Continuum hypothesis is true – properties
defined at infinitesimal points (density,
pressure, temperature, etc.)
Which fluids are ideal?
• Water – can be turbulent (waterfall not ideal,
ideal in a slow moving river)
• Air – compressible (piston not ideal, ideal in a
laminar wind)
• Honey – viscous fluid such that drag forces
can’t be neglected (Not usually ideal)
• Blood – pulsatile flow, filled with proteins/cells
(ideal in large arteries or veins, not capillaries)
What happens if fluid is not ideal?
Poiseuille's Law
• When frictional forces dominate velocity decreases
– Viscous fluids
– Small Diameters
Ideal – larger diameters
v
2p

With Friction – small diameters
r p
v
8L
2
Fluid Dynamics: Equation of Continuity
Equation of Continuity
• For an ideal fluid flowing in a pipe, the volume
flow rate through the pipe is constant
V
 Av  constant
t
A1v1  A2v2
Narrower section
Larger speed
Wider section
Smaller speed
Example: Water out of faucet
• Why does the stream of water flowing from a
faucet often get more narrow as the water
falls?
Gravity accelerates water so
velocity increases. If velocity
goes up, then area goes down…
http://thegoldenspiral.org/wp-content/uploads/2008/10/faucet_waterglass.jpg
Example: Arterial branching
• An artery branches into two smaller arteries,
each with half the diameter of the first. What
is the velocity in the smaller artery compared
to the larger artery?
A)
B)
C)
D)
Half
Same
Twice
Four times
http://cardiovascres.oxfordjournals.org/content/65/3/619/F4.small.gif
Fluid Dynamics: Bernoulli’s Equation
Bernoulli’s Equation
• For an ideal fluid flowing in a pipe, pressure in
the pipe is related to the velocity and height
of fluid
1 2
1 2
p1  gh1  v1  p2  gh2  v2
2
2
Discussion: Two sheets in the wind?
• What happens if I take two sheets of paper,
separate them by 1” and blow between them?
A) sheets will move apart
B) sheets will come together
C) sheets will stay at same spots
http://www.practicalphysics.org/imageLibrary/jpeg273/735.jpg
Main Points
• Buoyant force
FB   fluidVobject underwater g
• Ideal fluid is incompressible, laminar,
nonviscous, and irrotational
• Equation of continuity
• Bernoulli’s Equation
Av  constant
1 2
p  gh  v  constant
2